Question

As part of a research program for a new cholesterol drug, a pharmaceutical company would like to investigate the relationship between the ages and LDL (low-density lipoprotein) cholesterol of men. The accompanying data set shows the ages and LDL cholesterol levels of seven randomly selected men. Construct a 95% prediction interval to estimate the LDL cholesterol level of a 37 -year-old man.

Age Cholesterol

24 144

36 175

27 159

33 200

42 160

31 141

42 205

Answer 1

We will investigate the relationship by doing a simple linear regression using "age" as independent variable and "cholesterol" as dependent variable. Then we will find the prediction interval using the least square estimate. We will do all the necessary calculations in R. I will also attach the code to conduct the analysis.

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R-code :

age<-c(24,36,27,33,42,31,42) # importing age variable
cholst<-c(144,175,159,200,160,141,205) # importing cholesterol variable

fit<-lm(cholst~age) # Run simple linear regression
summary(fit) # summary of fit

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Output:

> summary(fit)

Call:

lm(formula = cholst ~ age)

Residuals:

1 2 3 4 5 6 7

-5.2732 0.8156 3.4990 32.0434 -26.6400 -22.8047 18.3600

Coefficients:

Estimate Std. Error t value Pr(>|t|)  

(Intercept) 99.451 46.096 2.157 0.0835 .

age 2.076 1.349 1.539 0.1843  

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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 22.95 on 5 degrees of freedom

Multiple R-squared: 0.3215, Adjusted R-squared: 0.1859

F-statistic: 2.37 on 1 and 5 DF, p-value: 0.1843

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Now we will attach the code to find the 95% prediction interval.

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new<-data.frame(age=37) # Creating a new data frame with only variable age as 37
predict(fit, new, interval = "predict") # finding the prediction interval

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Output:

> predict(fit, new, interval = "predict")
fit lwr upr
1 176.2604 112.0725 240.4482

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So, the 95% confidence interval is given by (112.0725, 240.4482)